Math Modeling: Our Favorite Ready-To-Use Problems That Encourage Communication and Collaboration

Lauren ShareshianGreta Mills

Cheryl Gann

http://tinyurl.com/ModelingNCTM2019

Introducing Modeling - Setting the Stage● Existing problems can be opened up to help create a modeling

mindset in your students.● These problems provide a great entry into mathematizing

real-world problems without being overwhelming.● The best introductory problems can be revisited and refined

throughout the course.

http://tinyurl.com/ModelingNCTM2019

After Hurricane Irma hit West Palm Beach in September 2017, Oxbridge’s Buildings and Grounds Department needed to order a replacement flag for the flagpole behind the school.

Standard flag sizing recommends that the length of the flag is at least 1/4 the height of the flagpole, and the width of the flag is approximately 2/3 the length of the flag.

In addition, flag protocol requires that the flag should never touch the ground, even when flying at half-staff.

Example: The Flagpole Problem

Example: The Flagpole ProblemUsing only standard measuring tools, determine the height of the flagpole behind Oxbridge Academy, and use your results to give the minimum and maximum dimensions of the replacement flag.

This introductory example sets the stage for further exploration of indirect measurement...

Example: The Olympic RingsDuring the 2012 Summer

Olympics, photographer Luke

MacGregor (Reuters) took

these photos of the moon

behind the Olympic Rings:

Example: The Olympic RingsHow far was the

photographer from the

bridge?

Example: Population EstimationPortland City Government is wondering how many people typically hang out at Waterfront Park on a sunny day. They take an aerial photo and look at a Google map of the area. Provide city officials with an estimate for the number of people in the park.

Even mini-modeling activities get all kids involved

● Simpler assumptions…

● To the more complex...

Even mini-modeling activities increase understanding● MANY students used the perimeter instead of area formula● Tons of area/perimeter book problems hadn’t made the concept stick● They needed a real world example of when to use which formula

The Bridge Program

• Students who have not historically been successful at traditional algebraic content can flourish during the modeling process.

• Students invited to participate in the NCSSM summer bridge program often have differences in preparation that could make it harder to experience success at NCSSM.

• Modeling is introduced to the students on the first day of the summer bridge program and incorporated throughout the two week math course.

The Bridge Program

• Some summer bridge students expressed their unease and frustration as they first approached the problem and explained how solving the problem gave them a great sense of accomplishment:

• “I learned to persevere and create a solution ‘that worked’ even though it wasn’t the ‘best answer’”.

• “I will keep this lesson with me and apply it to my classes in the following years as well as the rest of my life.”

• “If I'm looking to work in a STEM field one day, it is crucial to understand the applications of the different things we learn.”

Beginning to Model - Grouping StudentsWhen forming groups, some of the questions that a teacher may consider:

● Should I give students input on group formation?

● What do I do with a student who other kids don’t want to work with?

● What are some considerations in how I form groups?

● Should the groups be homogeneous or heterogeneous?

What are useful skills to spread out?● Big thinkers● Detail oriented kids● Strong coders & statisticians● Clear presentation slide makers● Good public speakers● Good time managers● The kid who will immediately opens up a Google Doc to keep everyone

organized and delegate

Heterogeneous vs. Homogeneous Groups● Heterogeneous groups are great when there are a variety of skills needed.

● Homogeneous groups are good when you don’t want the most advanced student to dominate within a group.

Homogeneous Example: The Midge Problem

Heterogenous Example: RollercoastersProblem: Create an objective roller coaster ranking system given a spreadsheet of data about each rollercoaster.

Question: What to do with the missing data?

The computer scientist dropped all missing values

The most advanced math student used machine learning to fill in the missing values

The most resourceful student used the basics“After doing a lot of internet research, I saw that a video of Montana Rusa in Ecuador was similar to that of Manta in San Diego, so the ratio of height of the two was used to calculate the speed of the Montana Rusa in Ecuador.”

All contributions are usefulWhile the machine learning student research steepest gradient descent, his partner pondered the more qualitative questions of:

“In addition to not being as long or as fast as a standard roller coaster, water rides cannot have inversions. Can we rank water rides using the same metric?”

“Few people want to spend excessive amounts of money for airplane tickets around the world. Thus, distance to the park should be taken into our metric.”

Beginning to Model - Expectations & ScaffoldingWhen transitioning from smaller doses of modeling into more extensive problems, the teacher can help students gain proficiency in mathematical modeling.

This can be achieved by sharing information about the modeling process, communicating expectations for the work and outcomes, and providing scaffolding to help students who are new to modeling.

Modeling Process Rubric - Sample Rubric 1

Scaffolding Example - Foraging for FoodBees can carry around 15 mg of pollen, but flowers don’t produce that much, so bees have to flit from flower to flower to collect enough pollen to fill their sacs before returning to the hive. In this problem you will explore the relationship between the availability of pollen and a bee’s optimal foraging strategy.

Scaffolding Example - Foraging for FoodHow this can be used to provide scaffolding:

● Students work in small groups on parts of the problem then throughout we come back together to share ideas and another small task is given.

● This small group to full class discussion is repeated throughout the work on the problem.

● Student groups are then assigned portions of the paper to draft.● We then discuss edits to the paper and students are given a final draft of the paper as

an example for future writing assignments.

Example - Driving for GasA friend tells you she buys her gas at a station several miles off your normal route where the prices are cheaper. Would it be more economical for you to drive the extra distance for the less expensive gas than to purchase gas along your route?

Image from xkcd.com

Identifying Variables - Gas Example • The price of gas at each station.

• The extra miles to the other station.

• The amount of gas I need to buy.

• The fuel efficiency of my car.

Ask an easier question first or create a specific example

Station A is on your normal route and sells gas for $2.25 a gallon, while Station B is 5 miles off your route sells gas for $2.10 a gallon. Your car gets 32 miles per gallon, and your friend’s car only gets 15 miles per gallon. Should either of you travel the extra distance to buy gas at Station B?

Identifying Variables and Making Assumptions• Variables in Textbook Problem: all quantities are given

• Variable in Modeling Problem: which quantities should we include?○ Play around with it first!○ Ask an easier question first.○ Create a specific example.

• Assumptions in Modeling Problem: finding the right balance○ Are we assuming so much that it oversimplifies the problem? Or assuming too little

and we can’t proceed○ Justify why the assumptions you are making are reasonable.○ Save early work - it might be useful for future steps.○ Add complexity step by step.

Iteration allows for differentiation• Some groups remain on first iteration while more advanced groups refine

their solution○ In the population problem, did you get one estimate? Now find

upper and lower bounds.

○ Did you use a single rectangle to approximate area? Now use smaller rectangles or more intricate shapes.

Iteration in the gas problem• Did you oversimplify? Add more complexity step by step.

Opportunities for DifferentiationThe modeling process provides many opportunities for differentiation, within a specific class or across multiple classes / levels.

Modeling problems have multiples entry points for students from early Algebra 2 through calculus, using a variety of tools from spreadsheets to differential equations.

The following example comes from the 2014 HiMCM contest and can be modified or extended.

Example: The Next PlagueIn a small island village almost half of its 300 inhabitants are showing similar symptoms of illness. In the past week, 15 of the “infected” have died. Your modeling team works for the International World Health Organization.

Determine and classify the type and severity of the spread of the disease.

The role of technology• Students use whatever mathematical tools they have.

• Simple to advanced use of technology and mathematical content.

• From spreadsheets and Desmos...

Technology Ebola Example• To using SageMath...

Technology Ebola Example• To using Netlogo...

Technology allows for differentiation• Students use whatever math they are most comfortable with.

• Leads to a variety of solutions, even among students in the same class.

•The next set of student slides all comes from the same modeling class.

Example: The Art Gallery ProblemAn art gallery needs to hold 50 paintings. Security cameras rotate at opposite ends of the room. The outer walls and portable walls have particular measurements, and the layout should be aesthetically pleasing. Where should you place the paintings in order to maximize security?

Differentiation – Art Gallery #1• Low-tech but effective (cardboard box and pencil and paper)

Differentiation – Art Gallery #2•Another group made such detailed trigonometric diagrams that you could

put them in a precalculus textbook!

Differentiation - Art Gallery #3: CalculusAnother group used calculus concepts to quantify security

Differentiation – Art Gallery #4: Geogebra

Differentiation – Art Gallery #5: Google Sketchup

Analyzing and Assessing the Final Solution•Be able to communicate WHY your solution makes sense.

○ Is it practical? ○ Is it reasonable? ○ How confident are you?

•Reflect on your mistakes

•Be aware of your model’s strengths and weaknesses• Sensitivity analysis

○ Where does the model fall apart?○ What are the extremes?

Assessing solutions: Middle School ExampleLow quality justification....

High quality justification...

Assessing solutions: High School Example● Google Maps!

Assessing solutions: High School Example● Error Analysis

Grading The Final Product

Self Evaluation Questions● What strengths did you bring to the group?● Which tactics (graphical, numerical, or analytical) provided you with a

greater insight into the problem?● What was the greatest moment in your work, the greatest source of pride,

or the greatest challenge?● What specific contributions did you make to the work?

Group/Problem Evaluation Questions:● How well did you work together as a group?● How well do you feel your group used class time in working on this?● What difficulties did you encounter in your work as a group?● Would you recommend this problem to your peers? What tips would you

give them as they set to work?

Ensuring that everyone contributes***********Greta insert collaboration rubric****

Talk about splitting the points amongst the kids

Ensuring Inequities in Work are Addressed

“What role do you typically play in your group?”Most students are flexible“I think the role changes depending on what group I'm in. I normally contribute to where the group needs me to be.”

“The roles really change depending on the topics and our workload. It also heavily depends on the capabilities of my teammates.”

“The roles differ depending on what group I'm in. Sometimes I may take the leadership role, other times I help with coding/equation formation, and I also work on presentation slides.”

Some tend towards one role“I am usually the coding person, but I'd like to think that I can contribute to all aspects of group work.”

“I have consistently found myself in the position of being very macro-level and proposing ways that my team can go about solving a problem.”

“For some reason, I was always given the task of divvying up the work.”

“I tend to be the presentation slide person, and I particularly focus on taking on and researching the assumptions of our group. That said, I still do plenty of math and am looking to diversify my roles in the future.”

Most students report an equal division of labor

Sometimes the group fails...The most advanced student used techniques that no one else understood and didn’t delegate or communicate.

“I get the concept of my model, but I don't understand it fully to the point where I can comfortably modify or do calculations on my own. The group dynamic was a challenge, since Kelly was way ahead of us. I would like to be paired with someone more on my level next time.”

But they learn from their mistakes...“I think this time our group cooperated really well and the work is divided evenly. I really enjoyed working with my group members, and I hope I can continue to distribute work evenly in the future so that everyone can get the part they want to work on.”

Student Feedback“I very much enjoyed the collaborative nature of everything. It was good practice for work as a team moving forward into college and beyond..”

“I really enjoyed the group projects as it was fun hearing other people’s ideas and coming up with the best solution together. I hope to take away teamwork and the general skill of coming up with creative ideas of solving a problem that has multiple ways of solving.”

“What I enjoyed the most is how each of my groups broke down the problems we got together. I learned a way to find a starting point for problems that I had no idea how to approach. I think this problem solving skill is really important to academics and life.”

Our Favorite Modeling Problems Can Be Found At...● Our Presentation Problems: http://tinyurl.com/ModelingNCTM2019

● HIMCM (November competition)● M3 Challenge (March competition)● GAIMME REPORT● Robert Kaplinsky

Thanks!

● Lauren Shareshian, Oregon Episcopal School, shareshianl@oes.edu● Greta Mills, Oxbridge Academy, Gmills@oapb.org● Cheryl Gann, North Carolina School of Science and Math, gann@ncssm.edu